Maximizing Revenue with Dynamic Cloud Pricing: The Infinite Horizon Case

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1 Maximizing Reenue with Dynamic Cloud Pricing: The Infinite Horizon Case Hong Xu, Baochun Li Department of Electrical and Computer Engineering Uniersity of Toronto Abstract We study the infinite horizon dynamic pricing problem for an infrastructure cloud proider in the emerging cloud computing paradigm. The cloud proider, such as Amazon, proides computing capacity in the form of irtual instances and charges customers a time-arying price for the period they use the instances. The proider s problem is then to find an optimal pricing policy, in face of stochastic demand arrials and departures, so that the aerage expected reenue is maximized in the long run. We adopt a reenue management framework to tackle the problem. Optimality conditions and structural results are obtained for our stochastic formulation, which yield insights on the optimal pricing strategy. Numerical results erify our analysis and reeal additional properties of optimal pricing policies for the infinite horizon case. I. INTRODUCTION Cloud computing has recently attracted much attention from both industry and academia. Beyond technological adances, cloud computing also holds promises to change the economic landscape of computing. Pricing of the cloud resources is both a fundamental component of the cloud economy and a crucial system parameter for the cloud operator, because it directly impacts customer usage pattern and the utilization leel of the infrastructure. Static pricing remains the dominant form of pricing today. Howeer, it is intuitie to adopt a dynamic pricing policy in order to strategically influence demand in order to better utilize unused cloud capacity, and to generate more reenue. Dynamic pricing emerges as an attractie strategy to better cope with unpredictable customer demand. A recent such example is Amazon, whose elastic computing cloud platform EC2) ] introduced a spot pricing feature for its instances, with the spot price dynamically updated to match supply and demand 2] as in Fig.. Gien the flexibility to change the price on the spot, a natural question is, what is the optimal dynamic pricing policy that a cloud proider such as Amazon EC2 can adopt, in terms of maximizing its expected reenue amid fluctuating demand? On one hand, a cloud proider has incentie to price to the present, i.e. to set price as high as possible to extract more profit from current customers. On the other hand, doing so increases the risk of negatiely affecting future demand especially from low aluation customers. An important obseration in cloud computing is that computing resources, such as CPU cycles and bandwidth, are inherently perishable: if at some point in time they are not utilized they are of no alue. That is, we should also consider pricing to the future. It is non-triial to balance the intrinsic tradeoff with perishable capacity and stochastic demand. Spot price $USD) Jan 2 Jan 2 Jan 22 Fig.. The history of spot prices on Amazon EC2 s large linux instances API name m.large) in US-West- aailability zone, from Jan 2 2 to Jan Data are obtained ia the Amazon EC2 API. In trying to address this fundamental challenge, we adopt a reenue management framework from economics that deals with the problem of selling perishable resources, such as airline seats and hotel reserations, in order to maximize the expected reenue from a population of price sensitie customers 3]. Dynamic pricing has become an actie field of the reenue management literature, with successful real-world applications in trael and fashion industries 4], 5]. In our preious work 6], we addressed the problem in a finite horizon setting. In this work we extend the analysis to consider the infinite horizon setting, where the optimal price is only a function of the system utilization and not a function of time. The objectie is to maximize the aerage expected reenue rate in the limit as time goes to infinity. Our contributions in this paper are two-fold. First, we present an infinite horizon stochastic dynamic program for the reenue maximization problem in the cloud, with stochastic demand arrials and departures. We characterize optimality conditions for the problem and proe important structural results, such as the monotonicity of optimal price and relatie rewards. Second, we conduct numerical studies to erify our analysis. The results also reeal seeral interesting obserations regarding the interplay between the degree of demand dynamics and the optimal pricing policy. Dynamic pricing is more important and rewarding when the expected dynamics is significant compared

2 2 with the system capacity. The remainder of the paper is structured as follows. In Sec. II we summarize related work. In Sec. III we present our formulation, and in Sec. IV we present analysis of the stochastic reenue maximization problem in the infinite horizon setting. Sec. V presents numerical results with different demand dynamics. Finally, Sec. VI concludes the paper. II. RELATED WORK Research on pricing in data communication networks started arguably from the work of Kelly 7], while there is certainly much prior work on pricing in telecommunication networks 8]. The book by Courcoubetis and Weber 9] proides a useful oeriew of the field. ] studies congestion-based pricing in cellular networks, and shows that static pricing achiees good performance in general. This result is extended in ] and 2]. The focus of these works is on the use of pricing to preent congestion in large network asymptotics, and the problem formulation assumes a single charge at the time a call is admitted, while our formulation differs by charging on the usage time basis. Our study is also related to another line of research on Internet bandwidth pricing for ISPs. The main theme here is to understand the tradeoff among resource allocation, performance, and social welfare implied by using flat-rate or usagebased pricing schemes, or a combination of the two called Paris Metro Pricing 3] 5]. In essence, a static price is used no matter if it is charged on a flat-rate or a usage basis, whereas we consider dynamic prices that ary oer time. There hae been some recent studies on pricing of cloud resources. 6] argues for the importance of pricing in the cloud computing context for distributed systems design. 7] proposes a computationally efficient pricing scheme based on mechanism design, and 8] adopts a genetic algorithm to iteratiely optimize the pricing policy. The most related work to ours is 9] where a pricing strategy is deeloped by soling an optimization problem for a cloud cache. These approaches are primarily of a one-shot nature without considering the effect of pricing on future demand and reenue. A. Model and Assumptions III. PROBLEM FORMULATION We start by introducing our model and assumptions. We focus on an infrastructure cloud proider that sells its computing resources packaged in the form of irtual machines, or instances, with a price that is dynamically changing oer time. We assume that the spot price p can take any alue from the interal, p max ]. Without loss of generality, we let p max = throughout the paper. We assume that the cloud operator can influence demand by arying its price p. Demand is determined by two independent Poisson processes, namely the arrial process that corresponds to the birth of new instances, and the departure process that models the death of existing ones when customers shut down the instances). Here we assume that a demand arrial function λp) determines the Poisson arrial rate number of new instances requested per time unit). Intuitiely, as price p increases, arrial rate strictly decreases, and thus λ p) <. The demand departure process is also a Poisson process, whose rate is modulated by the departure function µp). Clearly, µ p) > which means that customers are more likely to leae when spot price is higher. Price is a function of system state, and it is charged on a usage time basis. This means that, the proider collects p n from each of the actie instances eery unit of time, if there are n actie instances running in the system. n C and C denotes the system capacity. A pricing policy is a rule that determines which price should be adertised at any gien time as a function of the current state n. Since we are looking for a stationary pricing policy, it is not a function of time as in our preious work 6]. B. Formulation The reenue maximization problem can be formulated as follows. The goal of the cloud proider is to maximize the aerage reenue per unit of time in the infinite horizon. This quantity is denoted by J. We are interested in finding a pricing policy p = {p, p 2,..., p C } that achiees this goal, from the set P = {p p n, n} of all possible pricing policies. Under the aboe assumption, the system behaior follows the dynamics of a continuous-time birth-death Marko process, and explicit expression for the aerage profit J can be proided as follows. First, we denote the steady-state probability of state n gien a pricing policy p as π n p). The arrial and departure rates in any state gien p can be readily obtained, and the calculation of the steady-state probabilities π n p) is straightforward. Due to the Poisson Arrials See Time Aerages property, the aerage reenue rate is Jp) = C π n p) np n, ) n= since price is charged on a usage time basis in cloud computing. The proider s problem is to find a pricing policy p that maximizes the aerage reenue rate denoted by J. Equialently, J. = sup Jp). 2) p P This is a finite-state, continuous-time, aerage reward dynamic programming problem. Note that the set P is compact and all states communicate, so there always exists a policy with which we can eentually reach an arbitrary state n from any state n. The demand arrial and departure functions λp) and µp) are all continuous, and thus the transition rates and aerage reward rate J are continuous in the decision ariables p n. Moreoer, the reward rate and the expected holding time at each state n, and the total transition rate out of any state are all bounded functions of p. Under these assumptions, the standard DP theory applies 2], and there exists a stationary policy which is optimal.

3 3 A. Optimality Conditions IV. OPTIMAL DYNAMIC POLICIES 2) is a complex stochastic dynamic programming problem. To sole it, we can consider the so-called Bellman s equations since all the states in the Marko chain are recurrent. Bellman s equations are formulated for discrete-time Marko chains. Thus, we need to discretize our Marko chain by a procedure called uniformization, where the transition rates out of each state are normalized by the maximum possible transition rate, which in our case is gien by = max p λp) + µp). 3) The uniformized Marko chain for our problem is illustrated in Fig. 2. p ) p ) p ) p C ) 2 3 C- C µp ) Fig. 2. p ) µp ) µp 2 ) p C ) µp C ) µp C ) The uniformized Marko chain for our problem. µp C ) With the aboe setup, we can obtain the Bellman s equations of the following form: J + h = max np n + λp hn + ) + µp hn ) p n P + λp µp ) ] h. 4) for n =, 2,..., C. The first term in the right-hand side is the reenue rate at state n. The second, third, and fourth terms are contributions to the reenue if the next transition is an arrial, a departure, or a recurrence, respectiely. The unknowns in the aboe equations are h and J. J represents the optimal expected reenue per unit time as discussed, and h denotes the relatie reward in state n. In particular, consider an optimal pricing policy that attains the maximum in 4) for eery state n. If we follow this policy starting from state n or state n, the expectation of the difference in aerage rewards oer the infinite horizon is equal to hn ) h)/. Note that 4) holds only for n, C ]. To deal with cases when n = and C, i.e. when there is no demand departure and arrial, respectiely, we can simply let h ) =, and hc + ) = hc). The solution to the Bellman s equations can be computed using classical DP algorithms such as the policy iterations or relatie alue iteration 2]. The resulted prices that maximize the right-hand side of 4) at each state n constitutes the optimal pricing policy π. B. Structural Results Although we hae found the Bellman s equations and can numerically sole them, obtaining a closed form solution is quite difficult for arbitrary demand arrial and departure functions. Howeer, we are able to characterize seeral important structural properties of the optimal solution to the dynamic program 4). Our first result is the monotonicity of the relatie rewards. It corresponds to the fact that it is more desirable to hae more actie instances and thus a higher utilization of the capacity, as they lead to higher reenue rate in the long run een though they imply fewer reenue opportunities in the future. Theorem : Monotonicity of h. h hn ) for all n =, 2,..., C. Proof: The proof can be carried out in similar methods as used in the proof of Theorem in ]. Due to limited space the details are omitted. We can further show that the relatie rewards exhibit diminishing marginal returns with respect to utilization. Theorem 2: Concaity of h. h is concae in n for all n =,,..., C. Proof: It suffices to show that h hn ) hn + ) + h, n, C ]. 5) We proe by constructing a feasible pricing policy that satisfies the aboe inequality at equality. Suppose p is the optimal pricing policy that achiee the maximum in the right-hand side of 4) for each state. Consider two copies of the system, which we refer to as System A and B, respectiely. System A starts from state n and follows the optimal policy. System B starts from state n + and also follows the optimal policy. Now consider a third copy of the system, System C. It starts with state n, and at any point in time, it sets the price as half of the sum of System A s and B s prices. Let h denote the relatie reward obtained from this pricing policy. Now by construction of the policy and the definition of h, we hae h = hn ) + hn + )). 2 Since h corresponds to the optimal relatie reward, h h, and thus the proof. We can now proe our main result, which is the monotonicity of the optimal price at each state. Theorem 3: Price Monotonicity. p n p n for n =, 2,..., C. Proof: For conenience, let us denote g n = hn + ) h. From Theorem 2 we know that g n g n. Rearranging the terms in 4) we hae J = np λp n + g µp n g n. 6) By definition of the optimal price p n for state n, any other price cannot make the right-hand side of 6) larger, i.e. J np n + g n λp n ) µp g n ) n,

4 4 which gies us p n p n g n λp n ) λp ) + g n µp n ) µp n ) )] n Similarly we can write J = n )p λp n + g n ) n n )p λp n + g n p n p n n ) n ) g n 2 µp n ) g n 2 µp g n λp n ) λp ) + g n 2 µp n ) µp n ) )] Thus for the two inequalities to hold, we must hae g n λp n n ) λp ) + g n µp n ) µp n ) )] g n λp n ) λp ) + g n 2 µp n ) µp n ) )], which holds only when p n p n since λ p) <, µ p) >, and g n g n g n 2. Theorem 3 has natural economic interpretations. When the system is heaily loaded, it is in the interest of the proider to set a higher price to obtain a higher reenue from customers, as well as to discourage future demand in order to preent the system from oerloading. When the system is lightly utilized, the proider can afford to adopt a lower price to attract more customers. This key insight is also consistent with results in our preious work 6] and related work ]. V. NUMERICAL STUDIES In this section, we conduct numerical studies to ealuate the properties of the optimal pricing policy. The system capacity C is set to. This corresponds to a moderate-scale data center, such as a single aailability zone in Amazon EC2, with seeral tens of thousands of irtual instances according to anecdotal eidences 2]. We adopt the demand arrial and departure functions of the form λp) = a p 2 ) and µp) = bp 2, respectiely. In this case, the optimal price for each state n has a closed form solution p = n/2g n a+g n b), where g n = h n+ h n as seen in Sec. IV-B. A. Weak dynamics scenario We first consider a relatiely weak dynamics scenario, where the maximum expected demand arrials and departures a and b are orders of magnitude less than the system capacity C. We set one hour to be the unit of time, and assume that the cloud is expected to launch and close seeral hundreds of instances per hour in the weak dynamics scenario. Thus we set a and b to, which is much smaller than the system capacity C =. = in this case. The results are shown in Fig. 3. The optimal price increases with n number of actie instances) as seen from Fig. 3a). The relatie reward h n clearly grows with n as seen from Fig. 3b). These obserations alidate our analysis in Sec. IV-B. Note that the optimal price does not change much when n increases from 25 to, and is close to for the interal. This is due to the effect of small demand dynamics compared to een a lightly loaded system when n =.25C). The expected long-term reenue can be maximized without considering much about the future demand, i.e. setting price close to to obtain a higher reenue rate at present. To facilitate the understanding, Fig. 3c) shows a sample path of the optimal price, with the corresponding system state nt) starting from n = 5 and last for.25 unit time, i.e. 5 minutes. We can see that in this time period, p nt)) decreases only marginally from.982 to around.98, when nt) slowly decreases from 5 to about 493. Another obseration is that the relatie reward is almost linear in n as seen in Fig. 3a). This also can be explained by the small dynamics the system faces. Oer time, the system state n is not expected to change much as seen in Fig. 3c) and price p does not ary much with n, implying that the long-term reenue rate is proportional to n. When the system utilization is quite low below x = =.C), reenue generated from future demand becomes more important, and p is much lower and aries with n as in Fig. 3a). B. Strong dynamics scenario Now we examine the optimal pricing policy when the problem embraces a significant degree of demand dynamics. We let a and b equal to, which means the maximum number of new instances and anished instances) the cloud can expect is equal to its capacity. = in this case. Other parameters remain the same as in the preious experiment. The results are shown in Fig. 4. The optimal price and relatie rewards again exhibit monotonicity as expected from our analysis. Compared to the small dynamics case, the first obseration is that optimal price aries significantly with state n for the entire range of n, and increases much slower than it does in the small dynamics case in Fig. 3a). As seen from Fig. 4a), p 9) is much higher than p 5) which is in turn much higher than p ). The reason is that when the dynamics is strong, reenue from future demand is crucial in maximizing the long-term reenue rate. The operator cannot set a price close to in the hope of maximizing the present reenue rate, because doing so has detrimental effect on future reenue. The optimal policy thus is to slowly and steadily increase the price as n grows. This also explains the stronger concaity of h as seen in Fig. 4b), because the marginal benefit of increasing the utilization is diminishing, causing the reenue cure to be bent downwards. Therefore, dynamic pricing becomes more critical in a strong dynamics setting. We can expect the optimal price fluctuating with the utilization, since the number of instances in the system is expected to fluctuate quickly oer time. This is demonstrated in the sample path of the system state and price in Fig. 4a). Compared with Fig. 3a), the price grows from.3 to around.5 as n also grows. It is thus reasonable to conclude that dynamic pricing plays an important role in the strong dynamics setting.

5 p * a) Optimal price h b) Relatie reward p * nt)) nt) Time t hour) c) A sample path of nt) and the corresponding optimal price p nt)). Fig. 3. Numerical results with C =, λp) = p 2 ), µp) = p p * a) Optimal price h b) Relatie reward p * nt)). nt) Time t hour) c) A sample path of nt) and the corresponding optimal price p nt)). Fig. 4. Numerical results with C =, λp) = p 2 ), µp) = p 2, and N = 5 time interals. VI. CONCLUDING REMARKS In this paper, we presented an infinite horizon reenue maximization framework to tackle the dynamic pricing problem in an infrastructure cloud. The technical challenge compared to preious pricing work is that prices are charged on a usage time basis, and as a result the demand departure process has to be explicitly modelled. An aerage reward dynamic program is formulated for the infinite horizon case. Its optimality conditions and structural results on optimal pricing policies were presented. We showed that the relatie rewards as well as the optimal price exhibit monotonicity, which is resonant with preious results 6], ]. We also conducted numerical studies to erify the analysis, and illustrated the importance of dynamic pricing especially in the strong demand dynamics scenarios. REFERENCES ] Amazon EC2, 2] Amazon EC2 spot instances, 3] L. Weatherford and S. Bodily, A taxonomy and research oeriew of perishable-asset reenue management: Yield management, oerbooking, and pricing, Operations Research, ol. 4, no. 5, pp , September ] M. K. Geraghty and E. Johnson, Reenue management saes national car rental, Interfaces, ol. 27, no., pp. 7 27, ] B. Smith, R. J. Leimkuhler, and J. S. Darrow, Yield management at American Airlines, Interfaces, ol. 22, pp. 8 3, ] H. Xu and B. Li, Reenue maximization with dynamic cloud pricing, 2, in submission. 7] F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, Rate control for communication networks: Shadow prices, proportional fairness and stability, J. Operat. Res. Soc., ol. 49, no. 3, pp , March ] B. M. Mitchell and I. Vogelsang, Telecommunications Pricing: Theory and Practice. Combridge Uniersity Press, 99. 9] C. Courcoubetis and R. Weber, Pricing Communications Networks: Economics, Technology and Modelling. John Wiley & Sons, Ltd., 23. ] I. C. Paschalidis and J. N. Tsitsiklis, Congestion-dependent pricing of network serices, IEEE/ACM Trans. Netw., ol. 8, pp. 7 84, April 2. ] I. C. Paschalidis and Y. Liu, Pricing in multiserice loss networks: Static pricing, asymptotic optimality, and demand substitution effects, IEEE/ACM Trans. Netw., ol., pp , June 22. 2] X. Lin and N. Shroff, Simplification of Network Dynamics in Large Systems, IEEE/ACM Trans. Netw., ol. 3, pp , August 25. 3] A. Odlyzko, Paris Metro Pricing for the Internet, in Proc. ACM EC, ] X. R. Cao, H. X. Shen, R. Milito, and P. Wirth, Internet pricing with a game theoretical approach: Concepts and examples, IEEE/ACM Trans. Netw., ol., no. 2, pp , April 22. 5] C.-K. Chau, Q. Wang, and D.-M. Chiu, On the iability of Paris Metro Pricing for communication and serice networks, in Proc. IEEE INFOCOM, 2. 6] H. Wang, Q. Jing, R. Chen, B. He, Z. Qian, and L. Zhou, Distributed systems meet economics: Pricing in the cloud, in Proc. HotCloud : 2nd USENIX Conf. Hot Topics in Cloud Computing, 2. 7] M. Mihailescu and Y. M. Teo, On economic and computational-efficient resource pricing in large distributed systems, in Proc. th IEEE/ACM Intl. Symp. Cluster, Cloud and Grid Computing, 2. 8] M. Macías and J. Guitart, A genetic model for pricing in cloud computing markets, in Proc. 26th Symp. Applied Computing, 2. 9] V. Kantere, D. Dash, G. Francois, S. Kyriakopoulou, and A. Ailamaki, Optimal serice pricing for a cloud cache, IEEE Trans. Knowl. Data Eng., ol. 23, no. 9, pp , February 2. 2] D. P. Bertsekas, Dynamic Programming and Optimal Control. Academic Press, ] Amazon s EC2 Generating 22M+ Annually, blogpost, amazons-ec2-generating-22m-annually.